Concentration and Dilution

In clinical practice, nurses work with solutions every day — IV fluids, wound irrigations, topical preparations, and injectable medications. Understanding concentration tells you exactly how much active ingredient (solute) is dissolved in a given amount of solution. Getting this wrong can mean giving a patient too much or too little medication.

What Concentration Means

Concentration describes the amount of solute (drug or substance) dissolved in a total volume of solution. It can be expressed in several ways:

Percent Strength

Percent strength (weight/volume or w/v) means grams of solute per 100 mL of solution:

$\text{Percent Strength (w/v)} = \frac{\text{Grams of solute}}{\text{Total mL of solution}} \times 100$

Common Clinical Solutions

Example 1: How Much NaCl Is in a 1-Liter Bag of NS?

NS is 0.9%, meaning 0.9 g NaCl per 100 mL. A 1-liter bag contains 1000 mL.

$\text{NaCl} = \frac{0.9 \text{ g}}{100 \text{ mL}} \times 1000 \text{ mL} = 9 \text{ g}$

Answer: A 1 L bag of NS contains 9 grams of NaCl.

Ratio Strength

Ratio strength is written as 1:X, meaning 1 gram of solute in X mL of solution.

• Epinephrine 1:1,000 = 1 g per 1,000 mL = 1 mg/mL
• Epinephrine 1:10,000 = 1 g per 10,000 mL = 0.1 mg/mL

Safety alert — ratio expressions are obsolete for epinephrine. The ratio notations 1:1,000 and 1:10,000 were officially eliminated by FDA labeling requirements and USP standards beginning in 2016 because they caused fatal medication errors — clinicians confused the two concentrations, which differ tenfold. Epinephrine must now be labeled by mass concentration: 1 mg/mL (for IM/subcutaneous use, formerly 1:1,000) and 0.1 mg/mL (for IV use, formerly 1:10,000). You may still encounter the old ratio notation on exams, in older references, or on legacy stock in some facilities. Always verify the mg/mL concentration on the actual vial label before administering.

To convert ratio strength to mg/mL:

$\text{mg/mL} = \frac{1000 \text{ mg}}{\text{ratio denominator (mL)}}$

Example 2: Concentration of Epinephrine 1 mg/mL (Formerly 1:1,000)

Using the ratio-to-mg/mL conversion for reference:

$\text{mg/mL} = \frac{1000 \text{ mg}}{1000 \text{ mL}} = 1 \text{ mg/mL}$

Answer: Epinephrine at this concentration is 1 mg/mL. Modern labels state the concentration directly rather than using the legacy 1:1,000 ratio notation.

The Dilution Formula: $C_1 V_1 = C_2 V_2$

When you need to dilute a stronger (more concentrated) solution to create a weaker one, use the dilution formula:

$C_1 V_1 = C_2 V_2$

Where:

• $C_1$ = concentration of the stock (original) solution
• $V_1$ = volume of the stock solution you need
• $C_2$ = concentration of the desired (final) solution
• $V_2$ = volume of the desired solution

This formula works because the total amount of solute stays the same — you are only adding diluent (usually sterile water or NS).

Example 3: Diluting a Lidocaine Solution

Problem: You have Lidocaine 2% and need to prepare 50 mL of Lidocaine 1%. How much of the 2% solution do you need, and how much diluent do you add?

Step 1: Identify the variables.

• $C_1 = 2\%$, $V_1 = ?$
• $C_2 = 1\%$, $V_2 = 50 \text{ mL}$

Step 2: Solve for $V_1$.

$V_1 = \frac{C_2 \times V_2}{C_1} = \frac{1\% \times 50 \text{ mL}}{2\%} = 25 \text{ mL}$

Step 3: Calculate the diluent volume.

$\text{Diluent} = V_2 - V_1 = 50 - 25 = 25 \text{ mL}$

Answer: Mix 25 mL of Lidocaine 2% with 25 mL of diluent to make 50 mL of Lidocaine 1%.

Converting Between Concentration Formats

Nurses frequently need to convert between percent strength and mg/mL:

$1\% = \frac{1 \text{ g}}{100 \text{ mL}} = \frac{1000 \text{ mg}}{100 \text{ mL}} = 10 \text{ mg/mL}$

Quick rule: To convert percent to mg/mL, multiply by 10.

Common Mistakes to Avoid

1. Confusing percent with mg/mL. Remember that 1% = 10 mg/mL, not 1 mg/mL. This is one of the most common and dangerous errors in concentration math.
2. Mixing up ratio strength directions. When ratio notation is used, a larger denominator means a weaker solution — 1:1,000 (1 mg/mL) is ten times more concentrated than 1:10,000 (0.1 mg/mL). This exact confusion is why FDA/USP eliminated ratio expressions for epinephrine labeling.
3. Forgetting that $C_1 V_1 = C_2 V_2$ requires the same units. Both concentrations must be in the same unit (both as %, both as mg/mL, etc.) before you solve.
4. Calculating the diluent volume incorrectly. The diluent volume is $V_2 - V_1$, not $V_2$ alone. You are adding diluent to the stock volume to reach the total final volume.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Concentration tells you how much solute is in a given amount of solution
• Percent strength (w/v) means grams per 100 mL — to convert to mg/mL, multiply by 10
• Ratio strength (1:X) means 1 g of solute in X mL of solution — a larger denominator means a weaker solution
• The dilution formula $C_1 V_1 = C_2 V_2$ lets you calculate how to mix a desired concentration from a stock solution
• Always verify that both concentrations are in the same unit before applying the dilution formula
• When in doubt about a concentration, verify with the pharmacist before administering

Return to Math for Nurses for more topics in this section.